Blender material issue while snapping two objects snaps together
I think there are two reasons here for discontinuous smooth shading.


The normals are flipped on the right half ... CtrlF > F, Flip normals
The normals haven't got anything to interpolate to over the edges of the hemispheres.


To fix (2), retain an intact version of the sphere as a form, same topology, same location, and transfer the normals from it to the 2 halves, by giving them  Data Transfer modifiers, with the intact sphere as target, and the settings shown. You will have to set 'Autosmooth' on the hemispheres in their data tabs.